Answer :
a) The steady state level of output per effective worker is given by:
y* = 10k*^(1/2) = 100
The steady state level of investment per effective worker is given by:
i* = sy = 32
The steady state level of consumption per effective worker is given by:
c* = (1-s)y = 68
b) So the growth rate of total labor income is 6 percent (4 percent from population growth and 2 percent from productivity growth).
c) (i) At the steady state, the capital-output ratio is constant because both k and y are constant.
(ii) The real wage grows at the rate of technological progress, which is 2 percent in this case.
a) To find the steady state levels, we need to solve for the values of k, y, i, and c that satisfy the condition that the change in k over time is zero. At steady state, investment per effective worker equals depreciation per effective worker.
The production function is given by y = 10k^(1/2), so we can write the change in k over time as:
dk/dt = s*y - (n+δ)k
where s is the saving rate, n is the rate of population growth, and δ is the depreciation rate (which is 1/10 in this case, since capital lasts 10 years).
Setting dk/dt equal to zero, we get:
s*10k^(1/2) - (n+δ)k = 0
Solving for k, we get:
k* = [s*10^2/(n+δ)^2]
Substituting the values given, we get:
k* = [0.32*10^2/(0.04+0.1)^2] = 100
So the steady state level of capital per effective worker is 100.
The steady state level of output per effective worker is given by:
y* = 10k*^(1/2) = 100
The steady state level of investment per effective worker is given by:
i* = sy = 32
The steady state level of consumption per effective worker is given by:
c* = (1-s)y = 68
b) At the steady state, capital and labor each earn a constant share of the economy's income, so the growth rate of total capital income is equal to the growth rate of output per effective worker, which is 0.
The growth rate of total labor income is equal to the growth rate of the labor force, which is the sum of the growth rates of population and labor productivity (which is growing at a rate of 2 percent due to technological progress). So the growth rate of total labor income is 6 percent (4 percent from population growth and 2 percent from productivity growth).
c) (i) At the steady state, the capital-output ratio is constant because both k and y are constant.
(ii) The real wage grows at the rate of technological progress, which is 2 percent in this case.
To prove this, we can use the production function to write the real wage as:
w = (1/2)y/L
where L is the number of effective workers.
Taking the time derivative of w, we get:
dw/dt = (1/2)*(dy/dt)/L - (1/2)y(dL/dt)/L^2
At steady state, dy/dt and dL/dt are both zero, so we have:
dw/dt = 0.02
So the real wage grows at the rate of technological progress, which is 2 percent.
d) The Golden Rule level of capital per effective worker is the level that maximizes consumption per effective worker in the long run.
To find the Golden Rule level, we need to set the marginal product of capital equal to the depreciation rate:
d(y/k)/dk = δ
Taking the derivative of the production function with respect to k, we get:
dy/dk = 5k^(-1/2)
Setting this equal to δ and solving for k, we get:
k_GR = (25/δ^2) = 250
The Golden Rule level of capital per effective worker is 250.
At this level of capital per effective worker, output per effective worker is:
y_GR = 10k_GR^(1/2) = 50
The saving and investment per effective
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